Electrical impedance tomography using equipotential line method
(A new approach to Inverse Conductivity Problem) (A new approach to Inverse Conductivity Problem) (A new approach to Inverse Conductivity Problem) (A new approach to Inverse Conductivity Problem)
June-Yub Lee (Ewha), Ohin Kwon (Konkuk), Jeong-Rock Yoon (RPI)
Physical Background
(Forward) Conductivity Problem
Ω
∂
∂ =
= ∂
Ω
Ω
∈
on n g
OR u f
u
t s in
x u find
x x
given For
. . )
(
, ),
σ (
=0
∇
⋅
∇ σ u Inverse Conductivity Problem
Ω
∂
∂ =
= ∂ g on
n AND u
f u
given from
x Find σ ( )
Ω
) ∂
, ( f g
Impedance
~Voltage/Current
Tomography
Electrical Impedance Electrical Impedance Electrical Impedance
Electrical Impedance TomographyTomographyTomography (EIT)Tomography (EIT)(EIT)(EIT)
생체조직의 저항률 (20-100KHz)
10 100 1000 10000 100000
체액 혈장 혈액
근육 (횡
방향)
간 팔근
육 신경
조직 심장
근육
근육 (종
방향)
폐 지방 뼈 최소값
최대값
[Ω-cm]
경혈 및 경락의 전기적 특성
가 설
경혈과 경락 전기저항이 낮은 지점이다.
즉, 조직의 저항률이 낮다.
A model problem with heart and lung A model problem with heart and lung A model problem with heart and lung A model problem with heart and lung
Methods of EIT/ICP
n g u u
x u
fk k
Min∑ − ( ) where∇⋅ ∇ = 0, ∂∂ =
Objective σ 2 σ
σ
σ
voltage fi =
current gi =
) σ (x
• Single Measurement
⇒ minimum information
• Many Measurement
⇒ full information(?) on σ
• Infinite Measurement
⇒ mathematical theory
Numerical Obstacles for EIT/ICP
σ σ
σ
σ σ
σ σ σ
better ~ Find
, 0 Solve
Guss
, 0 where
) ( )
( Objective
)
2(
→
∂ =
= ∂
∇
⋅
∇
→
∂ =
= ∂
∇
⋅
∇
∑ − Ω n g u u
σ(x)
n g u u
x u x
f L
Min
• Strongly nonlinear:
• Ill-posed problem:
) (
)
( 2
2 Ω → ∈ ∂Ω
∈ L uσ L σ
σ
σ σ
σ ≈/ ~ → u ≈ u~
f u
n g
u u = → −
∂
= ∂
∇
⋅
∇ σ σ 0, σ Find better σ~ using σ Solve
σ 0 Regularity restrictiuon on σ condition
Stop u − f ≅ →
Ill-posed Nonlinear
• 50 iterations
EIT and MREIT
(Magenetic Resonace Electrical Impedance Tomography)
0.95 mA
Current Source Voltmeter
변수형 고속 영상복원 알고리즘
진단 변수 추출
심폐기능
소화기능
배뇨기능
뇌기능
비정상 조직 검출
기타 변수
실시간
실시간영상
영상모니터링
모니터링MREIT
고해상도 영상
적응형 요소융합 및 세분화
변수형 고속 알고리즘
초기해초기해 초기해초기해
(1)문제정의
(2)자속밀도의 영상화
MRCDI 및 MREIT의 원리
1 E
2 Ω E
∂Ω
(ρ, , ,V J B)
I I
n
3\Ω
L− L+
a
1 ( ) 0
( ) V ρ
∇ ∇ =
r
r
1 V on
n J
ρ ∂ = ∂Ω
∂
( ) 1 ( )
( ) V
= −ρ ∇
J r r
r
0
3
( ) ( ') ' '
4 '
J µ dv
π Ω
= × −
∫ r− r
B r J r
r r
3
0
\ 3
0
3
( ) ( ') ( ') ' '
4 '
( ') ( ') ' '
4 '
I
L
dv
I dl
µ π µ
π
±
±
Ω
= × −
−
= × −
−
∫
∫
r r
B r J r a r
r r r r r a r
r r
0
( ) 1 ( )
B
=µ ∇×
J r B r
1 E
E2
x y x y z
z 1
E
2 E
1 E
2 E x
y z Transversal Imaging
Slice
Three Sagittal Imaging Slices
Three Coronal Imaging Slices
전류밀도 영상 실험용 팬텀
자속밀도 및 전류밀도 영상
y
z
] Tesla [
y
z
] Tesla [
x
z
] Tesla [
x
z
] Tesla [
x
y
] Tesla [
x
y
] Tesla [
Bx By
Bz
] mA/mm
[ 2
x
y
] mA/mm [mA/mm2]
[ 2
x
y
|J|
0
1
= µ ∇×
J B
자속밀도 및 전류밀도 영상
J-Substitution Algorithm
Original
Conductivity
Reconstructed Conductivity
Current Density
Schematic Diagram for Equi-potential Line Method
Numerical Algorithm
Equipotential Line Method
Smooth conductivity distribution
Conductivity Result with noisy J
Phantom with 1%Add+10%Mul Noise
Artificially generated Human Body
1%Add+10%Mul Noise
Conclusions and further works
• Fast, stable and efficient
• Well-posed (error~noise)
• Better interpolation method
to handle discontinuous cases
• Noise handling for high conductivity ratio case where |J|~0
• 3D extension is trivial
But 3D data for J is expensive
Usage of Bz instead of J is better
